A conventional connection structure for connecting a circuit board or some other type of plate to a chassis or a motherboard includes a fixing element, such as a screw or a rivet, for connecting to between the circuit board/plate and the chassis/motherboard, so that the circuit board/plate is fixedly mounted on the chassis/motherboard. That is, the circuit board/plate is not freely movable once it is connected to the chassis/motherboard. Presently, however, in many cases of installing or mounting the circuit board/plate, the circuit board/plate must be slidable to a predetermined position and then be fastened to the chassis/motherboard. Therefore, the conventional connection structure can no longer meet the current connection requirement, and it is necessary to develop newer connection structures that must not be too complicate in design in order to be cost effective. Moreover, in many cases, the circuit board/plate must be installed in a vibration-proof manner. Therefore, the new connection structure should not only satisfy the above-described slidable, locating and fastening function, but also be vibration-proof or shock-absorbing. Therefore, it is an object of the present invention to develop a connection structure with fastening and shock-absorbing functions for connecting to between two plate members, such that one of the plate members can be slid relative to the other one to a predetermined position for connecting to the other plate in a vibration-proof manner.